A numerical method for computing time-periodic solutions in dissipative wave systems

TL;DR

This paper introduces a spectral, fast-converging numerical method for computing time-periodic solutions in dissipative wave systems, applicable to both stable and unstable solutions, demonstrated on several complex equations.

Contribution

The paper presents a novel numerical approach that determines unknown parameters and computes time-periodic solutions as boundary value problems using Newton-conjugate-gradient iterations.

Findings

Successfully applied to Kuramoto-Sivashinsky and Ginzburg-Landau equations.

Achieves spectral accuracy and fast convergence.

Applicable to both PDEs and ODE systems like Lorenz equations.

Abstract

A numerical method is proposed for computing time-periodic and relative time-periodic solutions in dissipative wave systems. In such solutions, the temporal period, and possibly other additional internal parameters such as the propagation constant, are unknown priori and need to be determined along with the solution itself. The main idea of the method is to first express those unknown parameters in terms of the solution through quasi-Rayleigh quotients, so that the resulting integro-differential equation is for the time-periodic solution only. Then this equation is computed in the combined spatiotemporal domain as a boundary value problem by Newton-conjugate-gradient iterations. The proposed method applies to both stable and unstable time-periodic solutions; its numerical accuracy is spectral; it is fast-converging; and its coding is short and simple. As numerical examples, this method is applied to the Kuramoto-Sivashinsky equation and the cubic-quintic Ginzburg-Landau equation, whose time-periodic or relative time-periodic solutions with spatially-periodic or spatially-localized profiles are computed. This method also applies to systems of ordinary differential equations, as is illustrated by its simple computation of periodic orbits in the Lorenz equations. MATLAB codes for all numerical examples are provided in appendices to illustrate the simple implementation of the proposed method.